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Russian Geology and Geophysics 49 (2008) 851–858
www.elsevier.com/locate/rgg
Compound modeling of Earth rotation and possible implications
for interaction of continents
V.P. Mel’nikov a, I.I. Smul’skii a,*, Ya.I. Smul’skii b
a
b
Institute of the Earth’s Cryosphere, Siberian Branch of the RAS, POB 1230, Tyumen’, 625000, Russia
Institute of Thermal Physics, Siberian Branch of the RAS, 1 prosp. Akad. Lavrent’eva, Novosibirsk, 630090, Russia
Received 28 March 2007; in revised form 11 March 2008; accepted 21 April 2008
Abstract
The theory of orbital climate forcing implies an approximate Earth rotation model which has never been checked against other methods.
We simulate the rotating Earth in compound models as a system of equal peripheral parts that orbit a central body and investigate the orbital
evolution as affected by the gravity pull from the Sun, the Moon, and the planets. The predicted nutation cycles agree well with estimates
by other methods. The model of periodic convergence and divergence of peripheral bodies may be useful to explain interaction of continents.
© 2008, IGM, Siberian Branch of the RAS. Published by Elsevier B.V. All rights reserved.
Keywords: Earth rotation; modeling; nutation; interaction of continents
Introduction and problem formulation
Modeling is a commonly used tool to look into complex
phenomena which are too hard to investigate otherwise. Such
are many global or astronomical issues, for instance, the
formation of protoplanets simulated in (Serbulenko, 1996), or
the behavior of the Earth’s spin axis over long periods of time
which is a challenging problem with many unknowns. We
address the dynamics of a rotating Earth to reconstruct the
evolution of its axis.
The amount of heat the Earth receives from the Sun
depends on its inclination relative to the incoming sunlight,
insolation time, and distance from the Sun. The position of
the Earth’s surface with respect to the Sun is controlled by
the position of the Earth’s orbit and spin axis changing under
the effect of the planets, the Moon, and the Sun. According
to Milankovitch’s theory (1930), the Earth’s surface insolation
at different latitudes depends on the orbital parameters (Fig. 1):
eccentricity, pericenter longitude ϕpγ = γB (position relative to
the moving equatorial plane), and especially on obliquity ε
(tilt of the Earth’s spin axis relative to its orbital plane). The
latter oscillates at period Tε = 41.1 kyr, which corresponds to
the major insolation cycle (Berger and Loutre, 1991). The
* Corresponding author.
E-mail address: [email protected] (I.I. Smul’skii)
obliquity cycle was derived from approximate analytical
equations of motion for an orbiting and rotating Earth.
Ever more precise solutions of the orbital motion equations
have been a subject of special recent research, and reliable
results are available for the evolution over 100 Myr. However,
the previous solutions of the Earth rotation problem were
schematic: Second-order differential equations were reduced
to first-order Poisson equations and the obtained estimate of
the obliquity cycle (Tε = 41.1 kyr) has never been checked
against other solutions. In this respect, there are some
questionable points. First, they are only second-order differential equations that can describe real rotational motion, and
recent numerical solutions of these equations (e.g., Smul’skii
and Sechenov, 2007) over kyr-scale time intervals gave
nutation cycles consistent with observed data. Second, the tilt
of the equatorial plane relative to the moving orbital plane is
unstable and subject to random change in first-order Poisson
solutions over long time intervals (Laskar, 1996; Laskar et al.,
2005), which supports our doubt on the reliability of the
equations themselves. Third, the obliquity ε found from
Poisson equations differs from that reported by ancient
astronomers (Newton, 1977).
The doubts on the available knowledge of ε evolution and
the complexity of the Earth’s rotation problem call for
alternative solutions. One such alternative is the subject of this
study.
Having explored the orbital motion and its evolution for
the Earth, the Moon and the planets of the Solar System over
1068-7971/$ - see front matter D 2008, IGM, Siberian Branch of the RAS. Published by Elsevier B.V. All rights reserved.
doi:10.1016/j.rgg.2008.04.003
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the lunar orbital axis is very similar to that of the terrestrial
spin axis N, as both precess around the moving axis S.
Inasmuch as the orbital axes of the planets and the Moon
have the same dynamics as the Earth’s rotation axis, it is
possible to formulate the Earth’s rotation problem by simulating a rotating Earth by several axisymmetrical bodies in an
equatorial plane. Their positions and velocities can be set
precisely with reference to the available exact solution for
interaction of such bodies (Grebenikov, 1998; Smul’skii, 1999,
2003b). The system will evolve under the effect of the planets,
the Sun, and the Moon, and the behavior of the compound
model of a rotating Earth will simulate the evolution of the
Earth’s spin axis.
Formulation of a compound model of Earth rotation
Fig. 1. Earth’s orbital elements and axes in fixed equatorial (xyz) and ecliptic
(xeyeze) coordinates. 1 — celestial sphere; 2 — Earth’s equatorial plane at T0;
3 — Earth’s orbital plane (fixed ecliptic plane) at T0; 4 — Earth’s equatorial
plane at T; 5 — Earth’s orbital plane (fixed ecliptic plane) at T; 5 — Earth’s
orbit relative to Sun and orbit of a peripheral body relative to Earth. Letters
denote unit vectors: N — Earth’s spin axis and orbital axis of a peripheral body;
S — Earth’s orbital axis; M — angular momentum of Solar System; γ0 — vernal
equinox at T0; B — pericenter longitude; ϕΩ = γ0γ2 — angular distance of orbital
ascending node; ϕp = γ2B — angular distance of pericenter; i — inclination of
orbital plane with respect to fixed equatorial plane; ϕpγ = γB – angular position
of pericenter in moving coordinates.
100 Myr (Mel’nikov et al. 2000; Mel’nikov and Smul’skii,
2004; Smul’skii, 2005), we obtained that the dynamics of the
Earth’s orbital plane (5 in Fig. 1) is controlled by the
precession of the Earth’s orbital axis S relative to the angular
momentum M of the Solar System with period T0E = 68.7 kyr
(Smul’skii, 2003a, 2005). The axis S rotates clockwise about
M, i.e., counter the orbital motion, while the angle between
S and M (nutation angle θS) varies from 0° to 2.9° at different
periods, being the greatest for the Earth’s orbital axis Tsn1 =
97.4 kyr.
According to astronomic evidence (Duboshin, 1976), the
Earth’s spin axis N (Fig. 1) precesses, also clockwise, about
the axis S, with period TprE = 25.7 kyr, and the angle ε
between N and S oscillates with different periods and
amplitudes. Thus, the Earth’s orbital axis S, as well as the
Earth’s rotation axis N, are subject to both nutation and
precession, but unlike N, the axis S rotates about the vector
M fixed in the space.
The same computation was applied to the Moon’s orbit for
time intervals of –2 Myr < T < 0 and –100 Myr < T < –98 Myr
and showed that the lunar orbital axis precesses about the
moving terrestrial orbital axis S at T = 18.6 years and has a
minor nutation with its cycle Tn = 0.47 year much shorter than
the precession cycle. Thus, as it turned out, the behavior of
The rotating Earth is simulated as a system of n bodies
with the equal masses m1 positioned axisymmetrically in the
equatorial plane around a central body of the mass m0 (Fig. 2).
The parameters of the compound model are found as follows.
The total mass of the peripheral bodies and the central body
equals the Earth’s mass ME:
ME = m1n + m0.
(1)
The peripheral bodies move along a circular orbit at the
angular velocity of the Earth’s rotation ωE, which, according
to the exact solution of the axisymmetrical problem (Smul’skii, 2003b), is given by
G (m0 + m1 fn)/a3 = ω2E,
(2)
where G is the acceleration due to gravity; a is the orbital
radius of n bodies, and the function fn depends on the number
of bodies n:
n
fn = 025 ∑
i=2
1
.
sin [(i − 1)π/n]
(3)
The Earth’s moments of inertia relative to the axis x (in
the equatorial plane) and to the system of the bodies are equal
being
JEx = 0.4ME R2Ep = 0.4m0R20 + 0.5m1na2,
(4)
where REp is the Earth’s polar radius and R0 is the radius of
the central body.
Equal are also their inertia moments with respect to the
axis z:
JEz = JEx /(1 − Ed) = 0.4m0R20 + m1na2,
(5)
where Ed = 3.2737752⋅10−3 is the Earth’s dynamic ellipticity.
The radiuses of the bodies R0 and R1 are inferred from
their masses and mean densities as
R0 = (3m0 /(4πρE))1/3, R1 = (3m1 /(4πρE))1/3,
where the Earth’s mean density is
(6)
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Fig. 2. Models of Earth rotation and model parameters at REe = 6.37816⋅106 m, ME = 5.9742⋅1024 kg; Trt = 2π/ω is orbital cycle of a peripheral body. Model 0:
m1⋅10−19 = 9.96⋅107 kg, a/REe = 4.45, Trt = 23.93 hr; model 1: m1⋅10−19 = 3.57 kg, a/REe = 6, Trt = 23.93 hr; model 2: m1⋅10−19 = 7.21 kg, a/REe = 6, Trt = 23.93 hr;
model 3: m1⋅10−19 = 0.54 kg, a/REe = 1, Trt = 1.408 hr.
ρE = 3ME/4πR3Ee,
(7)
REe being the Earth’s equatorial radius.
The system of four nonlinear algebraic equations (1), (2),
(4), and (5) defines four elements of the model: the masses
m0 and m1, the number n, and the orbital radius a. The system
was used to analyze the parameters in wide ranges, including
different n. However, there are two points of difficulty: (i) the
problem has no solution at some parameter values, and (ii)
the equations cannot fulfill exactly because the number of
bodies n is discrete. We chose a reasonable number of
peripheral bodies and used the following conditions instead of
(4) and (5). In model 1, the moment of inertia produced by
the peripheral bodies was assumed equal to the difference of
the Earth’s moments of inertia:
m1na2 = JEz − JEx.
(8)
In model 2 we equated the model moments of inertia given
by (4) and (5) with that of the Earth to obtain
2
0.5m1na
5/3
2/3
0.4m0 (3/(4πρE))
2
+ m1na
=
JEz − JEx
Jz
= Ed.
(9)
We integrated numerically the equations of motion (Mel’nikov and Smul’skii, 2004) of five bodies in the compound
model, the Moon, the nine planets of the Solar System, and
the Sun, i.e., sixteen bodies altogether, and investigated the
evolution of the two models (1 and 2). The resulting
precession cycle Tpr = 170 years of the bodies that orbit the
axis S turned to be times smaller than the precession of the
Earth’s axis N (TprE = 25.7 kyr). It was, however, possible to
increase the precession by reducing the model radius. Thus,
in model 3 the radius of the system was assumed to equal the
Earth’s equatorial radius a = REe and the bodies’ densities
were set twice the Earth’s density to avoid collisions:
ρ0 = 2ρE.
(10)
The angular velocity ω of an orbiting Earth was found
from (2):
ω2 = G⋅(m0 + m1 fn)/RE.
(11)
In addition to the three models of a rotating Earth, we tried
another one we called a zero model (Fig. 2). Its parameters
were found from (1)–(5) but at m0 = 0, i.e., without the central
body. Integration of the equations of motion predicted that the
bodies would go off the axial symmetry after two revolutions
and then converge rapidly to let the system collapse. The
reason is that the large mass of the peripheral bodies causes
large gravitation, which increases very fast when the system
looses its balance and the bodies converge by the pull from
the Moon and other celestial bodies. Therefore, we refused
that model and accepted the three others (Fig. 2).
The equations of motion for sixteen bodies were integrated
numerically at ∆t = 10−5 years for models 1 and 2 and at
∆t = 10−6 years for model 3. The integration results were
regularly saved in files, and the files were used to repeat the
calculation for an orbital cycle and 12 orbital elements of each
peripheral body relative to the Earth’s center (5 in Fig. 1),
including the ascending node longitude ϕΩ in the fixed
equatorial plane, the inclination of the orbital plane i with
respect to the equatorial plane, and the orbital eccentricity e.
The calculation was performed for points at different time
steps and over different time intervals up to 110 kyr, or
continuously through different quantities of peripheral body
revolutions, up to 20,000.
Results
Modeling bodies’ dynamics. Visualization of the integration results in various modes offered by the Galactica software
facilitates understanding the mechanism of the bodies’ motion
and choosing further steps of numerical experiments. Figure 3
shows nine positions of the peripheral bodies with respect to
the central body in model 3. The bodies move axisymmetrically before t = 0.328 year but after 0.339 year the symmetry
upsets and the bodies come into cyclic convergence-divergence motion: they converge the closest at 0.361 year and
then diverge to meet again at 0.094 year, and so on.
Similar cycles appear in other models as well. The
peripheral bodies in model 1 converge successively at 1.4, 2,
2.6, 3.2, 3.8, 4.7, 5, 5.6, 6.4, 7.1, 7.7, 8.2, and 8.9 years, i.e.,
the cycle is from 0.5 to 0.8 year, or 0.66 on average over
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Fig. 3. Dynamics of peripheral bodies in model 3 at different times (in years)
since T0.
15 years; in model 2 this average is 0.32 year. Model 2
assumes a twice larger mass of the peripheral bodies (Fig. 2)
which causes a four times greater gravitation and may be also
responsible for the twice shorter convergence cycle. In model
3, the peripheral bodies are spaced six times more closely and
their rotations are 17 times more frequent, thus making the
convergence cycle still shorter.
Although the bodies converge repeatedly, they never
approach the critical distances that would bring the system to
collapse. We integrated the equations of motion for the 110
kyr time interval in model 1 and 40 kyr in model 3 but found
no signs of possible collapse.
Orbital evolution of a peripheral body. After visual
examination of the bodies’ dynamics, we focused on the orbits
of peripheral bodies relative to the central body. See Fig. 4
for the evolution of three orbital elements (e, ϕΩ, i) in model 1
for 0...–0.5, 0...–5, and 0...–50 years. The angles ϕΩ and i
were considered in fixed equatorial coordinates with the
central body at the origin O (Fig. 1). The orbits of the
peripheral body were modeled continuously for all revolution
series rather than over time intervals.
The initial conditions were set for a circular orbit but the
eccentricity (e) was nonzero, though very small, already in the
first rotation (1 in Fig. 4), i.e., the orbit was not perfectly
circular. Then the eccentricity increased in a nonmonotonic
way, first with period T1 = 0.0714 year = 26.1 days, while
the tendency of time-dependent eccentricity growth extended
to longer intervals (0...–5 and 0...–50 years). See well
pronounced Te1 cycles at the 0...–5 years interval. Another
harmonic appeared in the last interval, with period T2 =
10 years, and e reached 6⋅10−4.
Judging by the ascending node ϕΩ = 4.3 rad ≈ 1.5π (1 in
Fig. 4), the orbit rotated about the y axis. At the beginning
ϕΩ varied aperiodically in a damped mode, and then grew
monotonously through 0...–5 and 0...–50 years.
The inclination i increased continuously through all intervals (Fig. 4) and then decreased again to zero having reached
the maximum imax = 0.815 rad (Fig. 5), at T = 170 year cycles.
The maximum imax is the double angle between the Earth’s
axes of rotation and orbit (N and S), i.e., 0.408 rad = 23.4°.
Then the angle ϕΩ increased on (Fig. 5) to reach 2π. The
2π value was subtracted from ϕΩ in the plots (at ϕΩ > 2π),
and the fall of i to zero corresponded to passage through 2π.
Thus, the ϕΩ growth continued until ϕΩ ≈ π/2 + 2π and then
broke at i = 0. This behavior of ϕΩ indicates clockwise rotation
of the orbit at Tpr, with almost six full rotations in 1000 year
(Fig. 5).
During the interval of 1000 years the eccentricity reached
e = 0.0015, with cycles (Te2 = 10 years) never longer than 10
years (Fig. 5).
The eccentricity e and the rotation cycle increased between
0 and 110 kyr at every 2 kyr while imax remained within
0.815 rad; the eccentricity likewise showed cyclic behavior
and reached the maximum of e = 0.0035.
Models 2 and 3 were explored in the same way and
likewise showed aperiodic change of ϕΩ in the beginning,
eccentricity cycles, and orbit rotation. The results for model 3
were, however, slightly different: the first eccentricity cycle
Fig. 4. Evolution of body orbital elements (e, ϕΩ, i) at each rotation in model 1
at time intervals: 1 — 0...–0.5 year; 2 — 0...–5 years; 3 — 0...–50 years. T are
Julian dates since 30.12.1949; step size corresponds to orbital period.
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V.P. Mel’nikov et al. / Russian Geology and Geophysics 49 (2008) 851–858
855
Fig. 5. Evolution of body orbital elements (e, ϕΩ, i) at each rotation in model 1.
Time interval is 1000 years, step size is 10 years.
was Te1 = 14.1 days, and 143 year cycles set up since some
time; the orbit rotation cycle was Tpr = 2604 years.
Orbital precession of a peripheral body. The orbital
elements i and ϕΩ can be used to find the projections of the
unit vector of the body’s axis N in the compound model onto
the fixed equatorial coordinate axes xyz (Fig. 1):
Nx = sin i⋅cos ϕΩ, Ny = − sin i⋅sin ϕΩ, Nz = cos i,
(12)
where N = √
N2x + N2y + N2z = 1.

The dashed circle in Fig. 1 marks the orbital planes of both
the Earth and the model bodies in order to simplify the image.
Let ϕE be the position of the ascending node of the Earth’s
moving orbit in the xyz coordinates and iE be the orbit
inclination relative to the fixed equatorial plane. Let the axis
zE in the coordinates xEyEzE related to the moving Earth’s
orbit be directed along the Earth’s orbit axis S and the axis
xE cross its ascending node γ in the fixed equatorial plane
(moving ecliptic coordinates). Then, the projections of the
body’s axis N onto the axes xEyEzE (Fig. 1) are
NxE = Nx cos ϕE + Ny sin ϕE,
(13)
NyE = Nx sin ϕE cos iE + Ny cos ϕE cos iE + Nz sin iE,
(14)
NzE = Nx sin ϕE sin iE + Ny cos ϕE sin iE + Nz cos iE.
(15)
The unit vector N for model 1 plotted in the plane xEyE
over two time intervals of 0...–1000 years and 0...–110 kyr
rotates around S (Fig. 6), and the data imaged relative to the
fixed Earth’s orbit are splitting. Testing the other models in
the same way indicated that the Earth’s spin axis N in this
compound model precesses about the axis S. The precession
cycles define the cycles of I and ϕΩ and are Tpr = 170 years
for models 1 and 2 and Tpr = 2604 years for model 3.
Spin axis nutation in the Earth’s compound model. The
angle between the moving rotation axes N and S in the Earth’s
compound model is
ε = arccos NzE.
(16)
Fig. 6. Precession of unit vector N of peripheral body (model 1) around moving
orbital axis S in different time intervals: 1 — 0...–1000 years; 2 — 0...–110 kyr;
3 — position of N during end of interval 0...–110 kyr.
We calculated the angles ε and studied their evolution for all
time intervals in the three models. For the shortest cycles we
took the difference ∆ε = ε − εs, where εs is the ε sliding
average obtained by averaging over an interval equal to the
double first cycle. The nutation cycles are shown for model 3
(4 in Fig. 7). In the 0...–1 year interval, the cycle is Tn1 =
13.9 days, which must be due to the lunar effect because Tn1
is half the Moon’s orbiting cycle. The nutation in the 0...–4
interval must be due to the Sun its cycle Tn2 = 0.5 year being
half the Earth’s orbiting cycle.
The period Tn3 in the 0...–50 year interval corresponds to
the rotation of lunar orbital nodes. This is the cycle at which
the Moon’s orbit oscillates with respect to the ecliptic plane
reaching its known “low” and “high” positions. Nutation at
Tn3 = 18.6 years characterizes the major lunar pull on the
Earth’s spin axis.
The longest nutation cycle of Tn4 = 2580 years appeared
in the interval 0...–6 kyr, where the nutation amplitudes
increased with period. Note that we additionally obtained
nutation cycles of 220 years, or less than Tn4 (4 in Fig. 7).
However, those cycles were rather apparent than real: that was
a longer-period image of the Tn3 = 18.6 year cycles of the
curve points sampled at every 20 years.
The maximum nutation cycles ε1 over the 0...–40 kyr
interval (scale T1, 4 in Fig. 7), with a time step of 1000 years,
almost never exceeded the cycles ε. Therefore, there were no
significant cycles larger than Tn4 = 2580 years in this interval.
The same modeling for the other cases (Table 1) showed
cycles like Tn1 to Tn3 but no cycles of Tn4 = 2580 years. We
tested the 0...–110 kyr interval in model 1 but found no
nutation with a significant amplitude and a period more than
Tn3. More tests were applied to look into the 10 kyr end of
the 0...–110 kyr interval where continuous rotations were at a
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Fig. 7. Nutation cycles of Earth’s rotation axis in different time intervals
predicted by compound model. ∆ε is deviation of nutation angle from sliding
average. See text for explanation.
cycle of 28 years. The nutation cycles were the same (Tn1,
Tn2, Tn3) as in Table 1. Thus, the ε nutation cycles were
invariable throughout the 110 kyr interval.
Discussion
Checking against other results. We developed a new
method to derive differential equations of the Earth’s rotational
motion and integrated them numerically to investigate the
effect of the Sun and the planets on the Earth (Smul’skii and
Sechenov, 2006). We found out (Smul’skii and Sechenov,
2007) that the cyclic effect of the celestial bodies on the
Earth’s rotation axis was due to the periodicity at which they
cross the equatorial plane, i.e., their orbiting half-periods. This
inference is perfectly consistent with the nutation cycles Tn1
and Tn2 related to the passage of the Moon and the Sun
through the Earth’s equator as obtained for the compound
model.
Note also a particular point concerning the reported results.
The neighbor ∆ε oscillations (1 in Fig. 7) are not identical but
repeat at Tn1. Thus, there may be another period 2Tn1 to
modulate the ∆ε cycles. The same is the case of Tn2 cycles
(2 in Fig. 7) if we represent them as the difference ∆ε. These
modulations at periods 2Tn1 = 1 month and 2Tn2 = 1 year are
due to the ellipticity of the lunar and solar orbits with respect
to the Earth. As a result, the two bodies cause different effects
on the Earth’s rotation when they pass through different sides
of the equatorial plane. These double cycles are commonly
used in processing observatory data.
For comparison, Table 1 lists major nutation periods and
amplitudes reported by Bretagnon et al. (1997) as complete
Earth’s rotation solutions for a short time interval. Our solution
for the case of the solar effect only gives a similar cycle of
Tn1 = 0.5 year and an amplitude of ∆ε2 = 0.27⋅10−5 rad. Thus,
the nutation cycles Tn2 and Tn3 (Table 1) derived from the
compound Earth rotation model coincide with those obtained
by solving the differential equations of rotational motion. The
amplitudes depend on model parameters and decrease proportionally to the orbital radius a; for the Earth they are an order
of magnitude smaller than in model 3.
As we wrote above, the theories of orbital climate forcing
invoke nutation cycles of Tε = 41.1 kyr, with amplitudes
∆ε = 1.7⋅10−2 rad = 0.98° for the first ∼400 kyr. The compound
Earth’s rotation models, however, predict much smaller amplitudes and shorter periods (Table 1).
Thus, the nutation cycles are the same in the compound
models and in the solutions of second-order differential
equations of the Earth’s rotation, and are conformable
with observation. Yet, the compound models shows no
Tε = 41.1 kyr cycles which were predicted by solving simplified first-order differential equations of rotational motion.
Possible mechanism of continent interaction. We studied
the evolution of the Earth’s axis using a model in which a
part of the Earth’s mass is divided between bodies distributed
Table 1
Precession and nutation parameters in three models of Earth’s rotation
Model
Tpr, years
Te1, days
Tn1, days
∆ε1⋅10−6, rad Tn2, years
∆ε2⋅10−5, rad Tn3, years
∆ε3⋅10−3, rad Tn4, years
∆ε3⋅10−3, rad
1
170
26
13.9
75
0.5
50
19.8
7.8
—
—
2
170
26
13.9
75
0.5
50
19.8
7.8
—
—
3
2604
27
13.9
5
0.5
3
18.6
0.39
2580
1.5
DERM
—
—
—
—
0.5
0.268
18.6
0.045
—
—
Note. ∆εi is nutation amplitude; DERM — according to solutions of differential equations of rotational motion.
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uniformly in its equatorial plane. Thus, it is, in a sense, a
model of “a dispersed Earth”. Note that a similar model can
be found in Isaac Newton’s works. As Laplace (1984) wrote,
Newton considered the Earth as a spheroid protuberant at the
equator, which is composed of a sphere and a meniscus around
it. The meniscus consists of orbiting particles, and the nodes
of their orbits are supposed to undergo retrograde motion
arising from the combined action of the Sun and the Moon.
From the connection of those bodies together there should
succeed a retrograde motion of the whole meniscus at the
intersection of the equator and the ecliptic, called the precession of the equinoxes; the retrograde motion is slow because
the meniscus divides it with the sphere to which it is attached.
Nobody before Newton could ever suspect the reason of the
great phenomenon depending on the compression of the Earth
and on the retrograde motion of meniscus nodes caused by
the Sun. These were two effects Newton was the first to
discover (Laplace, 1984). Thus, Newton likewise “decomposed” the Earth into separate bodies.
This model of Earth rotation allowed us to predict nutation
oscillations in addition to the precession of the Earth’s axis,
as well as cyclic changes of the distance between the
peripheral bodies. In model 3 the bodies behave as rhythmic
convergence and divergence (Fig. 3), which holds also in
models 1 and 2. Hence, convergence and divergence of bodies
is a general property of a rotating system. One can expect the
same motion for bodies lying on the Earth’s surface, which
actually corresponds to the case of model 3 with a = REe.
Everybody knows that the eastern borders of North and
South America mimic the contours of western Europe and
Africa. This striking match has been explained in various
hypotheses, and our modeling drives us to another explanation.
We think whether the multiple convergence cycles could shape
up the contours of the continents to adjust them to each other.
These motions should occur in the plane orthogonal to the
Earth’s rotation axis, i.e., in the way the bodies move in the
compound model.
The continents, treated as bodies rising above the Earth,
should experience the gravity pull from the Moon, the Sun,
and the planets, according to the prediction of our model.
However, unlike the peripheral bodies in our model, the
continents are not free but rooted deeply in the Earth’s interior,
so that their motion has to overcome viscous friction. As a
result, the convergence-divergence cycles can grow as long as
geological periods.
The bodies in the compound model approach each other as
close as one third of their orbit size. They never collide being
much smaller than the continents, which are subject to
collision. Thus, our model has implications for continent
interaction and may explain why the continent contours match
so strikingly.
Conclusions
We have justified and formulated a compound model of
Earth rotation and explored three models of a rotating Earth
857
for (i) the time corresponding to 20,000 continuous rotations
of a peripheral body and (ii) rotations at equal cycles within
a time span of 110 kyr.
The orbital axes precess (rotate against the orbital motion)
around the moving Earth’s orbit at cycles Tpr from 170 to
2604 years in different models.
The model Earth’s spin axis oscillates at periods of 14 days,
0.5 year, and 18.6 years in all models and also at 2508 years
in model 3. The first three nutation cycles are consistent with
solutions of the rotation equations and with observed data.
The three models predict no 41.1 kyr obliquity cycle
invoked in the existing theories of orbital climate forcing.
In their orbital motion, the peripheral bodies in the
compound model converge and diverge cyclically. Applied to
the case of continent interaction, these convergence-divergence cycles and the ensuing periodic collisions may explain
the perfect fit of continental margin geometries between two
Americas on the one side and Europe and Africa on the other
side.
All labor-consuming computation was run on an MCS1000
supercomputer at the Siberian Supercomputer Center, Siberian
Branch of the Russian Academy of Sciences (Novosibirsk),
where people were very helpful.
The paper profited much from constructive criticism and
editorial efforts of E.A. Grebenikov from the Computer Center
of the Russian Academy of Sciences (Moscow).
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